Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7519 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#7520 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1402 |
| commutator | C4⋊Q8 | C2×C4×C17 |
| Frattini, cosocle | #492 | C22×C17 |
| prime radical | C17×#127 | D4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C17 ◁ C2×C4×C17 ◁ C42×C17 ◁ C17×#492 ◁ G | |
156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,26,29,25)(21,30,31,24)(22,27)(23,28,33,32);(18,20,22,29)(19,23,27,30)(21,31,32,28)(24,25,33,26)374 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 508);